Advertisement

Cortical patterns and gamma genesis are modulated by reversal potentials and gap-junction diffusion

  • M.L. Steyn-Ross
  • D.A. Steyn-Ross
  • M.T. Wilson
  • J.W. Sleigh
Chapter
Part of the Springer Series in Computational Neuroscience book series (NEUROSCI, volume 4)

Abstract

In this chapter we describe a continuum model for the cortex that includes both axon-to-dendrite chemical synapses and direct neuron-to-neuron gap-junction diffusive synapses. The effectiveness of chemical synapses is determined by the voltage of the receiving dendrite V relative to its Nernst reversal potential \(V^{\rm rev}{}\). Here we explore two alternative strategies for incorporating dendritic reversal potentials, and uncover surprising differences in their stability properties and model dynamics. In the “slow-soma” variant, the \((V^{\rm rev} - V)\) weighting is applied after the input flux has been integrated at the dendrite, while for “fast-soma”, the weighting is applied directly to the input flux, prior to dendritic integration. For the slow-soma case, we find that–-provided the inhibitory diffusion (via gap-junctions) is sufficiently strong–-the cortex generates stationary Turing patterns of cortical activity. In contrast, the fast-soma destabilizes in favor of standing-wave spatial structures that oscillate at low-gamma frequency (\(\sim\)30-Hz); these spatial patterns broaden and weaken as diffusive coupling increases, and disappear altogether at moderate levels of diffusion. We speculate that the slow- and fast-soma models might correspond respectively to the idling and active modes of the cortex, with slow-soma patterns providing the default background state, and emergence of gamma oscillations in the fast-soma case signaling the transition into the cognitive state.

gap junctions cortical patterns gamma oscillation bifurcation Turing instability wave instability 

Notes

Acknowledgment

We thank Chris Rennie for helpful discussions on convolution formulations for the cortex. This research was supported by the Royal Society of New Zealand Marsden Fund, contract 07-UOW-037.

References

  1. 1.
    Alvarez-Maubecin, V., García-Hernández, F., Williams, J.T., Van Bockstaele, E.J.: Functional coupling between neurons and glia. J. Neurosci. 20, 4091–4098 (2000)PubMedGoogle Scholar
  2. 2.
    Bennett, M.V., Zukin, R.S.: Electrical coupling and neuronal synchronization in the mammalian brain. Neuron 41, 495–511 (2004)CrossRefPubMedGoogle Scholar
  3. 3.
    Bluhm, R.L., Miller, J., Lanius, R.A., Osuch, E.A., Boksman, K., Neufeld, R.W.J., Théberge, J., Schaefer, B., Williamson, P.: Spontaneous low frequency fluctuations in the BOLD signal in schizophrenic patients: Anomalies in the default network. Schizophrenia Bulletin 33(4), 1004–1012 (2007)CrossRefPubMedGoogle Scholar
  4. 4.
    Bressloff, P.C.: New mechanism for neural pattern formation. Phys. Rev. Lett. 76(24), 4644–4647 (1996), doi:10.1103/PhysRevLett.76.4644CrossRefGoogle Scholar
  5. 5.
    Coombes, S., Lord, G.J., Owen, M.R.: Waves and bumps in neuronal networks with axo-dendritic synaptic interactions. Physica D 178, 219–241 (2003), doi: 10.1016/S0167-2789(03)00002-2CrossRefGoogle Scholar
  6. 6.
    Ermentrout, G.B., Cowan, J.D.: Temporal oscillations in neuronal nets. Journal of Mathematical Biology 7, 265–280 (1979)CrossRefPubMedGoogle Scholar
  7. 7.
    Fox, M.D., Snyder, A.Z., Vincent, J.L., Corbetta, M., van Essen, D.C., Raichle, M.E.: The human brain is intrinsically organized into dynamic, anticorrelated functional networks. Proc. Natl. Acad. Sci. USA 102(27), 9673–9678 (2005), doi:10.1073/pnas.0504136102CrossRefGoogle Scholar
  8. 8.
    Fransson, P.: Human spontaneous low-frequency BOLD signal fluctuations: An fMRI investigation of the resting-state default mode of brain function hypothesis. Hum. Brain Mapp. 26, 15–29 (2005), doi:10.1002/hbm.20113CrossRefPubMedGoogle Scholar
  9. 9.
    Freeman, W.J.: Mass Action in the Nervous System. Academic Press, New York (1975)Google Scholar
  10. 10.
    Fukuda, T., Kosaka, T., Singer, W., Galuske, R.A.W.: Gap junctions among dendrites of cortical GABAergic neurons establish a dense and widespread intercolumnar network. J. Neurosci. 26, 3434–3443 (2006)CrossRefPubMedGoogle Scholar
  11. 11.
    Haken, H.: Brain Dynamics: Synchronization and Activity Patterns in Pulse-Coupled Neural Nets with Delays and Noise. Springer, Berlin (2002)Google Scholar
  12. 12.
    Hampson, E.C.G.M., Vaney, D.I., Weile, R.: Dopaminergic modulation of gap junction permeability between amacrine cells in mammalian retina. J. Neurosci. 12, 4911–4922 (1992)PubMedGoogle Scholar
  13. 13.
    Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.) 117, 500–544 (1952)Google Scholar
  14. 14.
    Hutt, A., Bestehorn, M., Wennekers, T.: Pattern formation in intracortical neuronal fields. Network: Computation in Neural Systems 14, 351–368 (2003)CrossRefGoogle Scholar
  15. 15.
    Laing, C.R., Troy, W.C., Gutkins, B., Ermentrout, G.B.: Multiple bumps in a neuronal model of working memory. SIAM J. Appl. Math. 63(1), 62–97 (2002), doi:10.1137/S0036139901389495CrossRefGoogle Scholar
  16. 16.
    Liley, D.T.J., Cadusch, P.J., Wright, J.J.: A continuum theory of electro-cortical activity. Neurocomputing 26–27, 795–800 (1999)CrossRefGoogle Scholar
  17. 17.
    Nadarajah, B., Thomaidou, D., Evans, W.H., Parnavelas, J.G.: Gap junctions in the adult cerebral cortex; Regional differences in their distribution and cellular expression of connexins. Journal of Comparative Neurology 376, 326–342 (1996)CrossRefPubMedGoogle Scholar
  18. 18.
    Nunez, P.L.: The brain wave function: A model for the EEG. Mathematical Biosciences 21, 279–297 (1974)CrossRefGoogle Scholar
  19. 19.
    Ouyang, L., Deng, W., Zeng, L., Li, D., Gao, Q., Jiang, L., Zou, L., Cui, L., Ma, X., Huang, X.: Decreased spontaneous low-frequency BOLD signal fluctuation in first-episode treatment-naive schizophrenia. Int. J. Magn. Reson. Imaging 1(1), 61–64 (2007)Google Scholar
  20. 20.
    Rennie, C.J., Wright, J.J., Robinson, P.A.: Mechanisms for cortical electrical activity and emergence of gamma rhythm. J. Theor. Biol. 205, 17–35 (2000)CrossRefPubMedGoogle Scholar
  21. 21.
    Robinson, P.A., Rennie, C.J., Wright, J.J.: Propagation and stability of waves of electrical activity in the cerebral cortex. Phys. Rev. E 56, 826–840 (1997)CrossRefGoogle Scholar
  22. 22.
    Rodriguez, E., George, N., Lachaux, J.P., Martinerie, J., Renault, B., Varela, F.J.: Perception’s shadow: long-distance synchronization of human brain activity. Nature 397, 430–433 (1999)CrossRefPubMedGoogle Scholar
  23. 23.
    Steyn-Ross, D.A., Steyn-Ross, M.L., Sleigh, J.W., Wilson, M.T., Gillies, I.P., Wright, J.J.: The sleep cycle modelled as a cortical phase transition. Journal of Biological Physics 31, 547–569 (2005)CrossRefGoogle Scholar
  24. 24.
    Steyn-Ross, M.L., Steyn-Ross, D.A., Sleigh, J.W.: Modelling general anaesthesia as a first-order phase transition in the cortex. Progress in Biophysics and Molecular Biology 85, 369–385 (2004)CrossRefPubMedGoogle Scholar
  25. 25.
    Steyn-Ross, M.L., Steyn-Ross, D.A., Sleigh, J.W., Liley, D.T.J.: Theoretical electroencephalogram stationary spectrum for a white-noise-driven cortex: Evidence for a general anesthetic-induced phase transition. Phys. Rev. E 60, 7299–7311 (1999)CrossRefGoogle Scholar
  26. 26.
    Steyn-Ross, M.L., Steyn-Ross, D.A., Wilson, M.T., Sleigh, J.W.: Gap junctions mediate large-scale Turing structures in a mean-field cortex driven by subcortical noise. Phys. Rev. E 76, 011916 (2007), doi:10.1103/PhysRevE.76.011916Google Scholar
  27. 27.
    Steyn-Ross, M.L., Steyn-Ross, D.A., Wilson, M.T., Sleigh, J.W.: Modeling brain activation patterns for the default and cognitive states. NeuroImage 45, 298–311 (2009), doi:10.1016/j.neuroimage.2008.11.036CrossRefPubMedGoogle Scholar
  28. 28.
    Steyn-Ross, M.L., Steyn-Ross, D.A., Wilson, M.T., Sleigh, J.W.: Interacting Turing and Hopf instabilities drive pattern formation in a noise-driven model cortex. In: R. Wang, F. Gu, E. Shen (eds.), Advances in Cognitive Neurodynamics ICCN 2007, chap. 40, pp. 227–232, Springer (2008)Google Scholar
  29. 29.
    Uhlhaas, P.J., Linden, D.E.J., Singer, W., Haenschel, C., Lindner, M., Maurer, K., Rodriguez, E.: Dysfunctional long-range coordination of neural activity during gestalt perception in schizophrenia. J. Neurosci. 26, 8168–8175 (2006)CrossRefPubMedGoogle Scholar
  30. 30.
    Uhlhaas, P.J., Singer, W.: Neural synchrony in brain disorders: Relevance for cognitive dysfunctions and pathophysiology. Neuron 52(1), 155–168 (2006), doi:10.1016/j.neuron.2006.09.020CrossRefPubMedGoogle Scholar
  31. 31.
    Wilson, H.R., Cowan, J.D.: A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13, 55–80 (1973)CrossRefPubMedGoogle Scholar
  32. 32.
    Wilson, M.T., Sleigh, J.W., Steyn-Ross, D.A., Steyn-Ross, M.L.: General anesthetic-induced seizures can be explained by a mean-field model of cortical dynamics. Anesthesiology 104, 588–593 (2006)CrossRefPubMedGoogle Scholar
  33. 33.
    Wilson, M.T., Steyn-Ross, D.A., Sleigh, J.W., Steyn-Ross, M.L., Wilcocks, L.C., Gillies, I.P.: The slow oscillation and K-complex in terms of a mean-field cortical model. Journal of Computational Neuroscience 21, 243–257 (2006)CrossRefPubMedGoogle Scholar
  34. 34.
    Wilson, M.T., Steyn-Ross, M.L., Steyn-Ross, D.A., Sleigh, J.W.: Predictions and simulations of cortical dynamics during natural sleep using a continuum approach. Phys. Rev. E 72, 051910 (2005)Google Scholar
  35. 35.
    Wright, J.J., Liley, D.T.J.: A millimetric-scale simulation of electrocortical wave dynamics based on anatomical estimates of cortical synaptic density. Network: Comput. Neural Syst. 5, 191–202 (1994), doi:10.1088/0954-898X/5/2/005CrossRefGoogle Scholar
  36. 36.
    Wright, J.J., Robinson, P.A., Rennie, C.J., Gordon, E., Bourke, P.D., Chapman, C.L., Hawthorn, N., Lees, G.J., Alexander, D.: Toward an integrated continuum model of cerebral dynamics: the cerebral rhythms, synchronous oscillation and cortical stability. BioSystems 73, 71–88 (2001)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • M.L. Steyn-Ross
    • 1
  • D.A. Steyn-Ross
    • 2
  • M.T. Wilson
    • 2
  • J.W. Sleigh
    • 2
  1. 1.Department of EngineeringUniversity of WaikatoHamiltonNew Zealand
  2. 2.Waikato Clinical SchoolUniversity of Auckland, Waikato HospitalHamiltonNew Zealand

Personalised recommendations